Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SEMICOMMUTATIVE PROPERTY ON NILPOTENT PRODUCTS

  • Kim, Nam Kyun (Faculty of Liberal Arts and Sciences Hanbat National University) ;
  • Kwak, Tai Keun (Department of Mathematics Daejin University) ;
  • Lee, Yang (Department of Mathematics Education Pusan National University)
  • Received : 2014.03.13
  • Published : 2014.11.01
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Abstract

The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are ${\sigma}$-rigid ${\delta}$-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where ${\sigma}$ is a ring endomorphism and ${\delta}$ is a ${\sigma}$-derivation.

Keywords

Acknowledgement

Supported by : NRF

References

  1. A. Alhevaz, M. Habibi, and A. Moussavi, On rings having McCoy-like conditions, Comm. Algebra 40 (2012), no. 4, 1195-1221. https://doi.org/10.1080/00927872.2010.548842
  2. A. Alhevaz, A. Moussavi, and M. Habibi, Nilpotent elements and skew polynomial rings, Algebra Colloq. 19 (2012), no. 1, 821-840. https://doi.org/10.1142/S1005386712000715
  3. D. D. Anderson and V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), no. 6, 2847-2852. https://doi.org/10.1080/00927879908826596
  4. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008), no. 8, 3128-3140. https://doi.org/10.1016/j.jalgebra.2008.01.019
  5. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470-473. https://doi.org/10.1017/S1446788700029190
  6. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970), 363-368. https://doi.org/10.1017/S0004972700042052
  7. W. Chen, On nil-semicommutative rings, Thai J. Math. 9 (2011), no. 1, 39-47.
  8. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noether- ian Rings, Cambridge University Press, Cambridge-New York-Port Chester-Melbourne- Sydney, 1989.
  9. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224. https://doi.org/10.1007/s10474-005-0191-1
  10. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226. https://doi.org/10.1016/S0022-4049(99)00020-1
  11. C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Rigid ideals and radicals of Ore extensions, Algebra Colloq. 12 (2005), no. 3, 399-412. https://doi.org/10.1142/S1005386705000374
  12. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative ring, Comm. Algebra 30 (2002), no. 2, 751-761. https://doi.org/10.1081/AGB-120013179
  13. S. U. Hwang, Y. C. Jeon, and Y. Lee, Structure and topological conditions of NI rings, J. Algebra 302 (2006), no. 1, 186-199. https://doi.org/10.1016/j.jalgebra.2006.02.032
  14. D. W. Jung, N. K. Kim, Y. Lee, and S. P. Yang, Nil-Armendariz rings and upper nilradicals, Internat. J. Algebra Comput. 22 (2012), 1250059, 13 pp. https://doi.org/10.1142/S0218196712500592
  15. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. https://doi.org/10.1006/jabr.1999.8017
  16. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300.
  17. T. K. Kwak and Y. Lee, Rings over which coefficients of nilpotent polynomials are nilpotent, Internat. J. Algebra Comput. 21 (2011), no. 5, 745-762. https://doi.org/10.1142/S0218196711006431
  18. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123. https://doi.org/10.1081/AGB-100002173
  19. A. R. Nasr-Isfahani, On skew triangular matrix ring, Comm. Algebra 39 (2011), no. 11, 4461-4469. https://doi.org/10.1080/00927872.2010.520177
  20. L. Motais de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, Proceedings of the 106th National Congress of Learned Societies (Perpignan, 1981), 71-73, Bib. Nat., Paris, 1982.
  21. L. Ouyang and H. Chen, On weak symmetric rings, Comm. Algebra 38 (2010), no. 2, 697-713. https://doi.org/10.1080/00927870902828702
  22. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17. https://doi.org/10.3792/pjaa.73.14
  23. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60. https://doi.org/10.1090/S0002-9947-1973-0338058-9
  24. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 233 (2000), no. 2, 427-436. https://doi.org/10.1006/jabr.2000.8451

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